Answer :
Sure, let's go step-by-step to find the distance from the center of the habitat to the point [tex]\((x, y)\)[/tex] using the distance formula.
1. Identify the Coordinates:
- For the center of the habitat, let's assume the coordinates [tex]\((x_1, y_1)\)[/tex].
- Here we assume the center is at the origin, so [tex]\((x_1, y_1) = (0, 0)\)[/tex].
2. Point Coordinates:
- The point whose distance we want to find is given as [tex]\((x, y)\)[/tex].
3. Distance Formula:
- The distance formula for the distance between two points [tex]\((x_1, y_1)\)[/tex] and [tex]\((x_2, y_2)\)[/tex] is:
[tex]\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \][/tex]
4. Substitute Coordinates:
- Plug in [tex]\((x_1, y_1) = (0, 0)\)[/tex] and [tex]\((x_2, y_2) = (x, y)\)[/tex] into the formula:
[tex]\[ d = \sqrt{(x - 0)^2 + (y - 0)^2} \][/tex]
5. Simplify the Expression:
- This simplifies to:
[tex]\[ d = \sqrt{x^2 + y^2} \][/tex]
So, the distance from the center of the habitat to the point [tex]\((x, y)\)[/tex] is:
[tex]\[ \sqrt{x^2 + y^2} \][/tex]
This expression [tex]\(\sqrt{x^2 + y^2}\)[/tex] represents the distance given in terms of [tex]\(x\)[/tex] and [tex]\(y\)[/tex].
1. Identify the Coordinates:
- For the center of the habitat, let's assume the coordinates [tex]\((x_1, y_1)\)[/tex].
- Here we assume the center is at the origin, so [tex]\((x_1, y_1) = (0, 0)\)[/tex].
2. Point Coordinates:
- The point whose distance we want to find is given as [tex]\((x, y)\)[/tex].
3. Distance Formula:
- The distance formula for the distance between two points [tex]\((x_1, y_1)\)[/tex] and [tex]\((x_2, y_2)\)[/tex] is:
[tex]\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \][/tex]
4. Substitute Coordinates:
- Plug in [tex]\((x_1, y_1) = (0, 0)\)[/tex] and [tex]\((x_2, y_2) = (x, y)\)[/tex] into the formula:
[tex]\[ d = \sqrt{(x - 0)^2 + (y - 0)^2} \][/tex]
5. Simplify the Expression:
- This simplifies to:
[tex]\[ d = \sqrt{x^2 + y^2} \][/tex]
So, the distance from the center of the habitat to the point [tex]\((x, y)\)[/tex] is:
[tex]\[ \sqrt{x^2 + y^2} \][/tex]
This expression [tex]\(\sqrt{x^2 + y^2}\)[/tex] represents the distance given in terms of [tex]\(x\)[/tex] and [tex]\(y\)[/tex].
